Abstract: A chain of Josephson junctions implements one of the simplest many-body
models undergoing a superconductor-insulator quantum phase transition between
states with zero and infinite resistance. This phenomenon is central to our
understanding of interacting bosons and fermions in one dimension. Apart from
zero resistance, the superconducting state is always accompanied by a
sound-like mode due to collective oscillations of the phase of the
complex-valued order parameter. Surprisingly little is known about the fate of
this mode upon entering the insulating state, where the order parameter
amplitude remains non zero, but the phase ordering is \melted" by quantum
fluctuations. Here we report momentum-resolved radio-frequency spectroscopy of
collective modes in nanofabricated chains of Al/AlOx/Al tunnel junctions. Our
key finding is that the GHz-frequency modes survive far into the insulating
regime, i.e. an insulator can superconduct AC currents. The insulating state
manifests itself by a weak decoherence of collective modes with an unusual
frequency dependence: longer wavelengths decohere faster, in fact suggesting
the absence of DC transport. Owing to an unprecedentedly large kinetic
inductance per unit length, the observed phase mode represents microwave
photons with a remarkably low speed of light (below 10^6 m/s) and high wave
impedance (above 23 kOhm). The latter exceeds the transition value for the Bose
glass insulator, expected in this system, by an order of magnitude, which
challenges theory to revisit the finite-energy condensate dynamics near the
transition. More generally, the high impedance translates into a fine structure
constant exceeding a unity, opening access to previously impossible regimes of
quantum electrodynamics.